Onp-adicL-functions and Zp-extensions of Certain Real Abelian Number Fields

AbstractLetkbe a real abelian number field with Galois groupΔandpan odd prime number. Denote byk∞the cyclotomic Zp-extension ofkwith Galois groupΓand byknthenth layer ofk∞/k. Assume that the order ofΔis prime topand thatpsplits completely ink/Q. In this article, we describe the order of theΓ-invariant part of theΨ-component of thep-Sylow subgroup of the ideal class group ofknfor sufficiently largen, in terms of a special value of thep-adicL-function associated toΨ, whereΨis an irreducible Qp-character ofΔ. This allows us to obtain an alternative formulation of Greenberg's criterion for the vanishing of theΨ-components of the cyclotomic Iwasawaλ- andμ-invariants ofkforp. We also compute some examples for cyclic cubic fields andp=5,7